Abstract. We propose an extragradient method with stepsizes bounded away from zero for stochastic variational inequalities requiring only pseudo-monotonicity. We provide convergence and complexity analysis, allowing for an unbounded feasible set, unbounded operator, non-uniform variance of the oracle and, also, we do not require any regularization. Alongside the stochastic approximation procedure, we iteratively reduce the variance of the stochastic error. Our method attains the optimal oracle complexity O(1/ǫ 2 ) (up to a logarithmic term) and a faster rate O(1/K) in terms of the mean (quadratic) natural residual and the D-gap function, where K is the number of iterations required for a given tolerance ǫ > 0. Such convergence rate represents an acceleration with respect to the stochastic error. The generated sequence also enjoys a new feature: the sequence is bounded in L p if the stochastic error has finite p-moment. Explicit estimates for the convergence rate, the oracle complexity and the p-moments are given depending on problem parameters and distance of the initial iterate to the solution set. Moreover, sharper constants are possible if the variance is uniform over the solution set or the feasible set. Our results provide new classes of stochastic variational inequalities for which a convergence rate of O(1/K) holds in terms of the mean-squared distance to the solution set. Our analysis includes the distributed solution of pseudo-monotone Cartesian variational inequalities under partial coordination of parameters between users of a network.

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We propose dynamic sampled stochastic approximated (DS-SA) extragradient methods for stochastic variational inequalities (SVI) that are robust with respect to an unknown Lipschitz constant L. We propose, to the best of our knowledge, the first provably convergent robust SA method with variance reduction, either for SVIs or stochastic optimization, assuming just an unbiased stochastic oracle and a large sample regime. This widens the applicability and improves, up to constants, the desired efficient acceleration of previous variance reduction methods, all of which still assume knowledge of L (and, hence, are not robust against its estimate). Precisely, compared to the iteration and oracle complexities of O(ǫ −2 ) of previous robust methods with a small stepsize policy, our robust method uses a DS-SA line search scheme obtaining the faster iteration complexity of O(ǫ −1 ) with oracle complexity of (ln L)O(dǫ −2 ) (up to log factors on ǫ −1 ) for a d-dimensional space. This matches, up to constants, the sample complexity of the sample average approximation estimator which does not assume additional problem information (such as L). Differently from previous robust methods for ill-conditioned problems, we allow an unbounded feasible set and an oracle with multiplicative noise (MN) whose variance is not necessarily uniformly bounded. These properties are appreciated in our complexity estimates which depend only on L and local variances or forth moments at solutions x * . The robustness and variance reduction properties of our DS-SA line search scheme come at the expense of nonmartingale-like dependencies (NMD) due to the needed inner statistical estimation of a lower bound for L. In order to handle a NMD and a MN, our proofs rely on a novel localization argument based on empirical process theory. Additionally, we propose a second provable convergent method for SVIs over the wider class of Hölder continuous operators without any knowledge of its endogenous parameters.

Artículo de publicación ISIForestry has contributed many problems to the Operations Research (OR) community. At the same time, OR has developed many models and solution methods for use in forestry. In this article, we describe the current status of research on the application of OR methods to forestry and a number of research challenges or open questions that we believe will be of interest to both researchers and practitioners. The areas covered include strategic, tactical and operational planning, firemanagement, conservation and the use ofORto address environmental concerns. The paper also considers more general methodological areas that are important to forestry including uncertainty,multiple objectives and hierarchical planning